Intermediate Value Theorem

If f is continuous on [a,b] and k is any number between f(a) and f(b), inclusive, then there is at least one number c in the interval [a,b] such that f(c) = k.

Intermediate Value Theorem

a

f(a)

bf(b)

k

c

Intermediate Value Theorem•an existence theorem; it

guarantees a number exists but doesn’t give a method for finding the number.•it says that a continuous function never takes on 2 values without taking on all the values between.

ExampleRyan was 20 inches long when born and 30 inches long when 9 months old. Since growth is continuous, there was a time between birth and 9 months when he was 25 inches long.

Why does the I. V. T. imply that an odd degree polynomial has at least one real root?

x

y

Do Not Assume the converse of the I.V.T.